On Applicability of Auxiliary System Approach to Detect Generalized Synchronization in Complex Network

Generalized synchronization is ubiquitous in nature. It is well known that the auxiliary system approach has been widely used to verify the presence of generalized synchronization. This approach was firstly proposed in a drive-response system, then extended to the bidirectionally coupled systems and complex networks. However, the well-known generalized auxiliary system method lacks a rigorous theoretical basis for its various applications. Two recent counterexamples indicate us the inapplicability of this method. Inspired by the counterexamples, we find that it is interesting to ask the following two fundamental questions: i) Why is the generalized auxiliary system approach unworkable in the networks with bidirectional couplings? ii) Are there any essential conditions for the applications of this approach? This technical note aims at establishing a rigorous theoretical basis for the applicability of auxiliary system approach. That is, the generalized auxiliary system approach is effective only if there does not exist any path from nodes to their driving neighbors (who drive these nodes) in a network. Several representative examples are also given to validate our theoretical results.